Advanced Thermodynamics Note 9 Vapor/Liquid Equilibrium: Introduction Lecturer: 郭修伯

Phase equilibrium Application Distillation, absorption, and extraction bring phases of different composition into contact. Both the extent of change and the rate of transfer depend on the departure of the system from equilibrium. Quantitative treatment of mass transfer the equilibrium T, P, and phase compositions must be known.

The nature of equilibrium A static condition in which no changes occur in the macroscopic properties of a system with time. At the microscopic level, conditions are not static. The average rate of passage of molecules is the same in both directions, and no net interphase transfer of material occurs. An isolated system consisting of liquid and vapor phases in intimate contact eventually reaches a final state wherein no tendency exists for change to occur within the system. Fixed temperature, pressure, and phase composition

Phase rule vs. Duhem’s theorem (The number of variables that is independently fixed in a system at equilibrium) = (the number of variables that characterize the intensive state of the system) - (the number of independent equations connecting the variable): Phase rule: Duhem’s rule: for any closed system formed initially from given masses of prescribed chemical species, the equilibrium state is completely determined when any two independent variables are fixed. Two ? When phase rule F = 1, at least one of the two variables must be extensive, and when F = 0, both must be extensive.

VLE: qualitative behavior When two chemical species: phase rule: the maximum value of F = 3 (π = 1), namely, P, T, and one mole fraction. All equilibrium states of the system can be represented in three-dimensional P-T-composition space. Fig 10.1

Pxy diagram at constant T Txy diagram at constant P Within this space, the states of pairs of phases coexisting at equilibrium define surfaces. The subcooled-liquid region lies above the upper surface; the superheated-vapor region lies below the under surface. UBHC1 and KAC2 represent the vapor pressure-vs.-T curves for pure species 1 and 2. C1 and C2 are the critical points of pure species 1 and 2. L is a bubble point and the upper surface is the bubblepoint surface. Line VL is an example of a tie line, which connects points representing phases in equilibrium. W is a dewpoint and the lower surface is the dewpoint surface. Pxy diagram at constant T Txy diagram at constant P PT diagram at constant composition

Fig 10.8 (a)(b), Negative departures from P-x1 linearity: strong liquid-phase inter-molecular attractions between unlike than between like pairs of molecules. Fig 10.8 (c)(d), Positive departures from P-x1 linearity: strong liquid-phase inter-molecular attractions between like than between unlike pairs of molecules. Fig 10.8 (b)(d), the “azeotrope”: the point where x1 = y1 the dewpoint and bubblepoint curves are tangent to the same horizontal line. The liquid does not change in composition as it evaporates. No separation of such a constant-boiling solution is possible by distillation. Fig 10.8

Fig 10.8

Fig 10.9

Fig 10.10

Simple models for VLE The simplest are Raoult’s law and Henry’s law. the vapor phase is an ideal gas (apply for low to moderate pressure) the liquid phase is an ideal solution (apply when the species that are chemically similar) although it provides a realistic description of actual behavior for a small class of systems, it is valid for any species present at a mole fraction approaching unity, provided that the vapor phase is an ideal gas.

For dewpoint calculation For bubblepoint calculation Binary system

Binary system acetonitrile (1)/nitromethane(2) conforms closely to Raoult’s law. Vapor pressures for the pure species are given by the following Antoine equations: (a) Prepare a graph showing P vs. x1 and P vs. y1 for a temperature of 75°C. (b) Prepare a graph showing t vs. x1 and t vs. y1 for a pressure of 70 kPa. (a) BUBL P At 75°C e.g. x1 = 0.6 At 75°C, a liquid mixture of 60 mol-% (1) and 40 mol-% (2) is in equilibrium with a vapor containing 74.83 mol-% (1) at pressure of 66.72 kPa. Fig. 10.11

Fig. 10.11

(b) BUBL T, having P = 70 kPa Select t t vs. x1 t vs. y1 Fig. 10.12

Henry’s law For a species present as a very dilute solute in the liquid phase, the partial pressure of the species in the vapor phase is directly proportional to its liquid-phase mole fraction: Table 10.1

Assuming that carbonated water contains only CO2 (species 1) and H2O (species 2) , determine the compositions of the vapor and liquid phases in a sealed can of “soda” and the pressure exerted on the can at 10°C. Henry’s constant for CO2 in water at 10°C is about 990 bar. Henry’s law for species 1: Raoult’s law for species 2: Assuming x1 = 0.01 assuming y1 = 1.0 Justified the assumption Justified the assumption

VLE modified Raoult’s law Account is taken of deviation from solution ideality in the liquid phase by a factor inserted into Raoult’s law: The activity coefficient, f (T, xi)

For the system methanol (1)/methyl acetate (2), the following equations provide a reasonable correlation for the activity coefficients: The Antoine equations provide vapor pressures: Calculate (a): P and {yi} for T = 318.15 K and x1 = 0.25 (b): P and {xi} for T = 318.15 K and y1 = 0.60 (c): T and {yi} for P = 101.33 kPa and x1 = 0.85 (d): T and {xi} for P = 101.33 kPa and y1 = 0.40 (e): the azeotropic pressure and the azeotropic composition for T = 318.15 K (a) for T = 318.15, and x1 = 0.25

(b): for T = 318.15 K and y1 = 0.60 An iterative process is applied, with Converges at: (c): for P = 101.33 kPa and x1 = 0.85 A iterative process is applied, with Converges at:

(d): for P = 101.33 kPa and y1 = 0.40 A iterative process is applied, with Converges at:

(e): the azeotropic pressure and the azeotropic composition for T = 318.15 K Define the relative volatility: Azeotrope Since α12 is a continuous function of x1: from 2.052 to 0.224, α12 = 1 at some point There exists the azeotrope!

VLE from K-value correlations A convenient measure, the K-value: the “lightness” of a constituent species, i.e., of its tendency to favor the vapor phase. The Raoult’s law: The modified Raoult’s law:

Fig 10.13

Fig 10.14

For a mixture of 10 mol-% methane, 20 mol-% ethane, and 70 mol-% propane at 50°F, determine: (a) the dewpoint pressure, (b) the bubblepoint pressure. The K-values are given by Fig. 10.13. (a) at its dewpoint, only an insignificant amount of liquid is present: (b) at bubblepoint, the system is almost completely condensed:

Flash calculations A liquid at a pressure equal to or greater than its bubblepoint pressure “flashes” or partially evaporates when the pressure is reduced, producing a two-phase system of vapor and liquid in equilibrium. Consider a system containing one mole of nonreacting chemical species: The moles of vapor The vapor mole fraction The moles of liquid The liquid mole fraction

The system acetone (1)/acetonitrile (2)/nitromethane(3) at 80°C and 110 kPa has the overall composition, z1 = 0.45, z2 = 0.35, z3 = 0.20, Assuming that Raoult’s law is appropriate to this system, determine L, V, {xi}, and {yi}. The vapor pressures of the pure species are given. Do a BUBL P calculation, with {zi} = {xi} : Do a DEW P calculation, with {zi} = {yi} : Since Pdew < P = 110 kPa < Pbubl, the system is in the two-phase region,